Multiplying fractions
🎯 In this topic you will
- Multiply two proper fractions
- Cancel common factors before multiplying fractions
- Estimate the answers to calculations
🧠 Key Words
- cancelling
- common factors
- to square
Show Definitions
- cancelling: Simplifying a fraction by dividing the numerator and denominator by a common factor.
- common factors: Numbers that divide exactly into two or more numbers.
- to square: To multiply a number by itself (e.g., $5^2 = 25$).
To find a fraction of a fraction, you multiply the fractions together.
The diagram shows a rectangle.
$\tfrac{1}{3}$ of the rectangle is blue.
You can see from the diagram that $\tfrac{1}{2}$ of $\tfrac{1}{3}$ = $\tfrac{1}{6}$.
This means that $\tfrac{1}{2} \text{ of } \tfrac{1}{3} = \tfrac{1}{2} \times \tfrac{1}{3} = \tfrac{1 \times 1}{2 \times 3} = \tfrac{1}{6}$.
So, when you multiply fractions, you multiply the numerators together and you multiply the denominators together.
❓ EXERCISES
1. Work out the following.
a. $\tfrac{1}{4} \times \tfrac{1}{2}$
b. $\tfrac{3}{4} \times \tfrac{1}{4}$
c. $\tfrac{2}{3} \times \tfrac{1}{5}$
d. $\tfrac{4}{5} \times \tfrac{2}{5}$
e. $\tfrac{3}{7} \times \tfrac{3}{4}$
f. $\tfrac{7}{9} \times \tfrac{2}{3}$
👀 Show answer
Answers (Q1):
- a. $\tfrac{1}{8}$
- b. $\tfrac{3}{16}$
- c. $\tfrac{2}{15}$
- d. $\tfrac{8}{25}$
- e. $\tfrac{9}{28}$
- f. $\tfrac{14}{27}$
2. Work out the following. Write each answer in its simplest form.
a. $\tfrac{3}{4} \times \tfrac{2}{5}$
b. $\tfrac{2}{3} \times \tfrac{3}{4}$
c. $\tfrac{4}{5} \times \tfrac{3}{8}$
d. $\tfrac{1}{4} \times \tfrac{8}{9}$
e. $\tfrac{3}{10} \times \tfrac{5}{6}$
f. $\tfrac{6}{11} \times \tfrac{1}{3}$
👀 Show answer
Answers (Q2):
- a. $\tfrac{3}{10}$
- b. $\tfrac{1}{2}$
- c. $\tfrac{3}{10}$
- d. $\tfrac{2}{9}$
- e. $\tfrac{1}{4}$
- f. $\tfrac{2}{11}$
3. Benji is making a sauce. This is the recipe he uses.
| Sauce (serves $4$ people) | |
|---|---|
| $\tfrac{2}{3}$ cup of cashew nuts | $2$ tablespoons of honey |
| $\tfrac{1}{3}$ cup of water | $\tfrac{1}{2}$ teaspoon of salt |
| $\tfrac{1}{4}$ cup of vinegar | |
Benji makes sauce for two people, so he multiplies all the amounts by $\tfrac{1}{2}$. Copy and complete the table, which shows the amount of each ingredient that Benji needs.
| Amount for $4$ people | Working | Amount for $2$ people |
|---|---|---|
| $\tfrac{2}{3}$ cup of cashew nuts | $\tfrac{1}{2} \times \tfrac{2}{3}$ | $\tfrac{1}{3}$ cup of cashew nuts |
| $\tfrac{1}{3}$ cup of water | $\tfrac{1}{2} \times \tfrac{1}{3}$ | $\tfrac{1}{6}$ cup of water |
| $\tfrac{1}{4}$ cup of vinegar | $\tfrac{1}{2} \times \tfrac{1}{4}$ | $\tfrac{1}{8}$ cup of vinegar |
| $2$ tablespoons of honey | $\tfrac{1}{2} \times 2$ | $1$ tablespoon of honey |
| $\tfrac{1}{2}$ teaspoon of salt | $\tfrac{1}{2} \times \tfrac{1}{2}$ | $\tfrac{1}{4}$ teaspoon of salt |
👀 Show answer
The completed amounts for $2$ people are:
- $\tfrac{1}{3}$ cup of cashew nuts
- $\tfrac{1}{6}$ cup of water
- $\tfrac{1}{8}$ cup of vinegar
- $1$ tablespoon of honey
- $\tfrac{1}{4}$ teaspoon of salt
4. Find the area of this rectangle.
🔎 Reasoning Tip
Area of a rectangle: Use the formula: \(\text{Area} = \text{length} \times \text{width}\).
👀 Show answer
Area of rectangle $= \text{length} \times \text{width}$
$= \tfrac{4}{9} \times \tfrac{1}{10}$
$= \tfrac{4}{90} = \tfrac{2}{45}$ m$^2$
5. Work out the area of this square.
👀 Show answer
Area of square $= \text{side}^2$
$= \left(\tfrac{3}{4}\right)^2$
$= \tfrac{9}{16}$ m$^2$
🧠 Think like a Mathematician
Question 6:
Look back at Question 5.
What methods can you use to square a fraction?
For example, what is:
$\left(\dfrac{3}{4}\right)^2$ ?
👀 show answer
To square a fraction, you square both the numerator and the denominator:
$\left(\dfrac{3}{4}\right)^2 = \dfrac{3^2}{4^2} = \dfrac{9}{16}$
✅ Answer: $\dfrac{9}{16}$
❓ EXERCISES
7. At a hotel, $\tfrac{5}{9}$ of the staff are employed part-time.
🔎 Reasoning Tip
Fraction of a fraction: In part c, you need to work out \( \tfrac{3}{7} \) of \( \tfrac{5}{9} \).
a. What fraction of the staff are not employed part-time?
b. Of the part-time members of staff, $\tfrac{3}{7}$ are men. What fraction of the part-time members of staff are women?
c. What fraction of the staff are men employed part-time?
d. What fraction of the staff are women employed part-time?
👀 Show answer
Answers (Q7):
- a. Not employed part-time $= 1 - \tfrac{5}{9} = \tfrac{4}{9}$
- b. Women part-time $= 1 - \tfrac{3}{7} = \tfrac{4}{7}$ of part-time staff
- c. Men part-time $= \tfrac{5}{9} \times \tfrac{3}{7} = \tfrac{15}{63} = \tfrac{5}{21}$ of staff
- d. Women part-time $= \tfrac{5}{9} \times \tfrac{4}{7} = \tfrac{20}{63}$ of staff
8. In a cinema, $\tfrac{3}{5}$ of the people watching the film are children. $\tfrac{3}{4}$ of the children are girls.
a. What fraction of the people watching the film are girls?
b. What fraction of the people watching the film are boys?
👀 Show answer
Answers (Q8):
- a. Girls = $\tfrac{3}{5} \times \tfrac{3}{4} = \tfrac{9}{20}$ of the people
- b. Boys = $\tfrac{3}{5} \times \tfrac{1}{4} = \tfrac{3}{20}$ of the people
🧠 Think like a Mathematician
Question 9:
Work out the answer to:
$\dfrac{6}{9} \times \dfrac{3}{12}$
What different methods could you use to work out the answer?
Discuss in pairs or in groups.
👀 show answer
Method 1 (Direct multiplication):
$\dfrac{6}{9} \times \dfrac{3}{12} = \dfrac{6 \times 3}{9 \times 12} = \dfrac{18}{108}$
Simplify: $\dfrac{18}{108} = \dfrac{1}{6}$
Method 2 (Simplify first):
$\dfrac{6}{9} = \dfrac{2}{3}, \quad \dfrac{3}{12} = \dfrac{1}{4}$
So, $\dfrac{2}{3} \times \dfrac{1}{4} = \dfrac{2}{12} = \dfrac{1}{6}$
✅ Answer: $\dfrac{1}{6}$
❓ EXERCISES
10. Arun says:
“When you multiply two proper fractions together, you will never get an answer bigger than $1$.”
Is Arun correct? Explain your answer. Look back at the questions you have completed in this exercise to help you explain.
👀 Show answer
Answer (Q10):
Yes, Arun is correct. A proper fraction is always less than $1$. When you multiply two numbers less than $1$, the product must also be less than $1$. For example, $\tfrac{3}{4} \times \tfrac{2}{5} = \tfrac{6}{20} = \tfrac{3}{10}$, which is less than $1$.
Therefore, multiplying two proper fractions will never give an answer bigger than $1$.
11. Samara uses the following method to estimate the answer to a multiplication.
Question: Work out $\tfrac{3}{4} \times \tfrac{1}{6}$.
Estimate: $\tfrac{3}{4}$ is greater than $\tfrac{1}{2}$ but less than $1$.
$\tfrac{1}{2}$ of $\tfrac{1}{6}$ is $\tfrac{1}{12}$, and $1 \times \tfrac{1}{6} = \tfrac{1}{6}$.
So, the answer must be greater than $\tfrac{1}{12}$ but smaller than $\tfrac{1}{6}$.
Accurate: $\tfrac{3}{4} \times \tfrac{1}{6} = \tfrac{3 \times 1}{4 \times 6} = \tfrac{3}{24} = \tfrac{1}{8}$.
$\tfrac{1}{8}$ is greater than $\tfrac{1}{12}$ but smaller than $\tfrac{1}{6}$.
For each of the following, use Samara’s method to first work out an estimate and then find the accurate answer.
a. $\tfrac{2}{3} \times \tfrac{1}{8}$
Use the fact that $\tfrac{2}{3}$ is greater than $\tfrac{1}{2}$ but less than $1$.
b. $\tfrac{2}{9} \times \tfrac{1}{4}$
Use the fact that $\tfrac{2}{9}$ is greater than $0$ but less than $\tfrac{1}{2}$.
c. $\tfrac{5}{8} \times \tfrac{4}{9}$
Use the fact that $\tfrac{5}{8}$ is greater than $\tfrac{1}{2}$ but less than $1$.
👀 Show answer
Answers (Q11):
- a. Estimate: between $\tfrac{1}{16}$ and $\tfrac{1}{8}$. Accurate: $\tfrac{2}{3} \times \tfrac{1}{8} = \tfrac{2}{24} = \tfrac{1}{12}$.
- b. Estimate: between $0$ and $\tfrac{1}{8}$. Accurate: $\tfrac{2}{9} \times \tfrac{1}{4} = \tfrac{2}{36} = \tfrac{1}{18}$.
- c. Estimate: between $\tfrac{2}{9}$ and $\tfrac{4}{9}$. Accurate: $\tfrac{5}{8} \times \tfrac{4}{9} = \tfrac{20}{72} = \tfrac{5}{18}$.
12. Copy this secret code box.
Work out the answer to each of the multiplications in the box.
Find the answer in the secret code box, then write the letter from the multiplications box above the answer.
For example: $\tfrac{1}{4} \times \tfrac{2}{3} = \tfrac{2}{12} = \tfrac{1}{6}$, so write **E** above $\tfrac{1}{6}$ in the secret code box.
What is the secret message?
👀 Show answer
Answer (Q12):
- E: $\tfrac{1}{4} \times \tfrac{2}{3} = \tfrac{1}{6}$
- U: $\tfrac{1}{5} \times \tfrac{1}{7} = \tfrac{1}{35}$
- L: $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}$
- I: $\tfrac{1}{5} \times \tfrac{2}{7} = \tfrac{2}{35}$
- H: $\tfrac{3}{4} \times \tfrac{6}{11} = \tfrac{18}{44} = \tfrac{9}{22}$
- S: $\tfrac{4}{9} \times \tfrac{5}{8} = \tfrac{20}{72} = \tfrac{5}{18}$
- F: $\tfrac{4}{5} \times \tfrac{7}{8} = \tfrac{28}{40} = \tfrac{7}{10}$
- A: $\tfrac{3}{5} \times \tfrac{2}{10} = \tfrac{6}{50} = \tfrac{3}{25}$
- N: $\tfrac{6}{7} \times \tfrac{2}{3} = \tfrac{12}{21} = \tfrac{4}{7}$
- T: $\tfrac{5}{7} \times \tfrac{2}{3} = \tfrac{10}{21}$
- M: $\tfrac{8}{9} \times \tfrac{4}{5} = \tfrac{32}{45}$
The secret message is: HELLO FUN MATHS.
13. The diagram shows a square joined to a rectangle. What is the total area of the shape?
👀 Show answer
Answer (Q13):
Square side $= \tfrac{3}{4}$ m, so area of square $= \left(\tfrac{3}{4}\right)^2 = \tfrac{9}{16}$ m$^2$.
Rectangle dimensions: length $= \tfrac{7}{8}$ m, width $= \tfrac{2}{5}$ m, so area $= \tfrac{7}{8} \times \tfrac{2}{5} = \tfrac{14}{40} = \tfrac{7}{20}$ m$^2$.
Total area $= \tfrac{9}{16} \tfrac{7}{20} = \tfrac{45}{80} \tfrac{28}{80} = \tfrac{73}{80}$ m$^2$.
🍬 Learning Bridge
Now that you understand how to multiply fractions by multiplying the numerators and denominators, you’re ready to make your work even easier. In the next section, you’ll learn how cancelling common factors before multiplying can simplify calculations, reduce large numbers, and help you reach answers in their simplest form more efficiently.
You already know how to multiply an integer by a fraction and a fraction by a fraction. You can complete multiplications more easily by cancelling common factors before you multiply.
❓ EXERCISES
14. Copy and complete these multiplications. Cancel common factors before multiplying.
a. $\tfrac{3}{4} \times 12$
b. $\tfrac{5}{7} \times 28$
c. $\tfrac{4}{5} \times 45$
d. $\tfrac{3}{8} \times 72$
👀 Show answer
Answers (Q1):
- a. $\tfrac{3}{4} \times 12 = 3 \times 3 = 9$
- b. $\tfrac{5}{7} \times 28 = 5 \times 4 = 20$
- c. $\tfrac{4}{5} \times 45 = 4 \times 9 = 36$
- d. $\tfrac{3}{8} \times 72 = 3 \times 9 = 27$
15. Copy and complete these multiplications. Cancel common factors before multiplying. Write each answer as a mixed number in its simplest form.
a. $\tfrac{3}{8} \times 36$
b. $\tfrac{4}{9} \times 39$
c. $\tfrac{5}{6} \times 8$
d. $\tfrac{7}{10} \times 45$
👀 Show answer
Answers (Q2):
- a. $\tfrac{3}{8} \times 36 = \tfrac{3}{2} \times 9 = \tfrac{27}{2} = 13 \tfrac{1}{2}$
- b. $\tfrac{4}{9} \times 39 = \tfrac{4}{3} \times 13 = \tfrac{52}{3} = 17 \tfrac{1}{3}$
- c. $\tfrac{5}{6} \times 8 = \tfrac{5}{3} \times 4 = \tfrac{20}{3} = 6 \tfrac{2}{3}$
- d. $\tfrac{7}{10} \times 45 = \tfrac{7}{2} \times 9 = \tfrac{63}{2} = 31 \tfrac{1}{2}$
🧠 Think like a Mathematician
Question 16:
Sofia is working out: $16 \times \dfrac{11}{24}$
Her method:
$16 \times \dfrac{11}{24} = \dfrac{16}{24} \times 11 = \dfrac{2}{3} \times 11 = \dfrac{22}{3} = 7\dfrac{1}{3}$
Sofia first got $7\dfrac{2}{6}$, then simplified further to $7\dfrac{1}{3}$. She wonders why her first answer wasn’t already in its lowest terms.
a) Why did Sofia have to simplify again, even after cancelling common factors first?
b) Write a solution for Sofia where she wouldn’t need to cancel at the end.
c) Fill in the missing words:
“When you cancel common factors before multiplying, if you cancel using the greatest common factor, your answer will always be in its simplest form.”
d) Discuss your answers with a partner.
👀 show answer
a) Sofia cancelled $16$ with $24$ first, leaving $\dfrac{4}{6}$, which simplifies later to $\dfrac{2}{3}$. Since she didn’t use the greatest common factor immediately, the fraction could still be simplified at the end.
b) A more direct method:
$16 \times \dfrac{11}{24} = \dfrac{16}{24} \times 11 = \dfrac{2}{3} \times 11 = \dfrac{22}{3} = 7\dfrac{1}{3}$ (already simplest).
c) The missing words: greatest common factor.
✅ Final Answer: $7\dfrac{1}{3}$
❓ EXERCISES
17. Work out these multiplications. Cancel common factors before multiplying.
a. $\tfrac{7}{11} \times 132$
b. $\tfrac{7}{9} \times 180$
c. $\tfrac{1}{12} \times 30$
d. $\tfrac{9}{14} \times 35$
👀 Show answer
Answers (Q4):
- a. $\tfrac{7}{11} \times 132 = 7 \times 12 = 84$
- b. $\tfrac{7}{9} \times 180 = 7 \times 20 = 140$
- c. $\tfrac{1}{12} \times 30 = \tfrac{30}{12} = \tfrac{5}{2} = 2 \tfrac{1}{2}$
- d. $\tfrac{9}{14} \times 35 = 9 \times \tfrac{5}{2} = \tfrac{45}{2} = 22 \tfrac{1}{2}$
18. Work out these multiplications. Cancel common factors before multiplying. Write each answer in its lowest terms. Two of them have been started for you.
a. $\tfrac{6}{7} \times \tfrac{5}{9}$
b. $\tfrac{3}{8} \times \tfrac{5}{6}$
c. $\tfrac{8}{9} \times \tfrac{3}{13}$
d. $\tfrac{8}{5} \times \tfrac{5}{12}$
e. $\tfrac{8}{21} \times \tfrac{9}{20}$
f. $\tfrac{2}{5} \times \tfrac{15}{16}$
👀 Show answer
Answers (Q5):
- a. $\tfrac{6}{7} \times \tfrac{5}{9} = \tfrac{30}{63} = \tfrac{10}{21}$
- b. $\tfrac{3}{8} \times \tfrac{5}{6} = \tfrac{15}{48} = \tfrac{5}{16}$
- c. $\tfrac{8}{9} \times \tfrac{3}{13} = \tfrac{24}{117} = \tfrac{8}{39}$
- d. $\tfrac{8}{5} \times \tfrac{5}{12} = \tfrac{40}{60} = \tfrac{2}{3}$
- e. $\tfrac{8}{21} \times \tfrac{9}{20} = \tfrac{72}{420} = \tfrac{6}{35}$
- f. $\tfrac{2}{5} \times \tfrac{15}{16} = \tfrac{30}{80} = \tfrac{3}{8}$
19. This is part of Razi’s homework. Razi works out the answer to the question by cancelling common factors before multiplying. He checks his answer is correct by cancelling common factors after multiplying.
Example:
I eat $\tfrac{1}{4}$ of a pizza. My brother eats $\tfrac{2}{3}$ of what is left. What fraction of the pizza does my brother eat?
Amount left $= 1 - \tfrac{1}{4} = \tfrac{3}{4}$
So $\tfrac{3}{4} \times \tfrac{2}{3} = \tfrac{1}{2}$
Check: $\tfrac{3}{4} \times \tfrac{2}{3} = \tfrac{6}{12} = \tfrac{1}{2}$
Use Razi’s method to work out and check the answers to these questions:
a. The guests at a party eat $\tfrac{5}{8}$ of a cake. Sam eats $\tfrac{1}{3}$ of what is left. What fraction of the cake does Sam eat?
b. The guests at a party eat $\tfrac{7}{10}$ of the rolls. Ed eats $\tfrac{5}{6}$ of what is left. What fraction of the rolls does Ed eat?
👀 Show answer
Answers (Q6):
- a. Left $= 1 - \tfrac{5}{8} = \tfrac{3}{8}$. Sam eats $\tfrac{1}{3} \times \tfrac{3}{8} = \tfrac{3}{24} = \tfrac{1}{8}$. ✅ Check: multiply directly = $\tfrac{1}{3} \times \tfrac{3}{8} = \tfrac{1}{8}$.
- b. Left $= 1 - \tfrac{7}{10} = \tfrac{3}{10}$. Ed eats $\tfrac{5}{6} \times \tfrac{3}{10} = \tfrac{15}{60} = \tfrac{1}{4}$. ✅ Check: multiply directly = $\tfrac{5}{6} \times \tfrac{3}{10} = \tfrac{1}{4}$.
20. Lewis uses this formula to work out the distance, in kilometres, his car will travel when he knows the average speed, in kilometres per hour, and the time in hours.
distance = average speed × time
Lewis thinks that if he drives for 50 minutes at an average speed of 220 kilometres per hour he will travel more than 180 km. Is Lewis correct? Explain your answer. Show all your working.
🔎 Reasoning Tip
Time conversion: Start by changing 50 minutes into a fraction of an hour, so you can use this fraction in the formula.
👀 Show answer
Answer (Q7):
Convert $50$ minutes into hours: $50 \div 60 = \tfrac{5}{6}$ hours.
Now use the formula:
$\text{distance} = 220 \times \tfrac{5}{6} = \tfrac{1100}{6} = 183.\overline{3}$ km
Since $183.\overline{3} \gt 180$, Lewis is correct. He will travel more than 180 km.
21. This is part of Mia’s classwork.
Example: Work out $2 \tfrac{1}{2} \times 2 \tfrac{4}{15}$
- Change to improper fractions: $\tfrac{5}{2} \times \tfrac{34}{15}$
- Cancel common factors: $\tfrac{5}{2} \times \tfrac{34}{15} = \tfrac{1}{1} \times \tfrac{17}{3}$
- Multiply: $\tfrac{17}{3}$
- Change to a mixed number: $5 \tfrac{2}{3}$
- Check with estimate: $5 \tfrac{2}{3} \approx 5$ ✅
Mia’s method to estimate the answer is to round one of the fractions to the nearest half and the other fraction to the nearest whole number. Use Mia’s method to estimate and to work out these multiplications. Write each answer as a mixed number in its simplest form.
a. $1 \tfrac{1}{2} \times 3 \tfrac{3}{5}$
b. $2 \tfrac{1}{4} \times 3 \tfrac{2}{3}$
c. $1 \tfrac{1}{8} \times 3 \tfrac{1}{6}$
d. $3 \tfrac{2}{3} \times 1 \tfrac{5}{22}$
e. $3 \tfrac{3}{4} \times 4 \tfrac{3}{5}$
f. $4 \tfrac{4}{7} \times 2 \tfrac{5}{16}$
👀 Show answer
Answers (Q8):
- a. $1 \tfrac{1}{2} = \tfrac{3}{2}, \; 3 \tfrac{3}{5} = \tfrac{18}{5}$ $\tfrac{3}{2} \times \tfrac{18}{5} = \tfrac{54}{10} = \tfrac{27}{5} = 5 \tfrac{2}{5}$
- b. $2 \tfrac{1}{4} = \tfrac{9}{4}, \; 3 \tfrac{2}{3} = \tfrac{11}{3}$ $\tfrac{9}{4} \times \tfrac{11}{3} = \tfrac{99}{12} = \tfrac{33}{4} = 8 \tfrac{1}{4}$
- c. $1 \tfrac{1}{8} = \tfrac{9}{8}, \; 3 \tfrac{1}{6} = \tfrac{19}{6}$ $\tfrac{9}{8} \times \tfrac{19}{6} = \tfrac{171}{48} = \tfrac{57}{16} = 3 \tfrac{9}{16}$
- d. $3 \tfrac{2}{3} = \tfrac{11}{3}, \; 1 \tfrac{5}{22} = \tfrac{27}{22}$ $\tfrac{11}{3} \times \tfrac{27}{22} = \tfrac{297}{66} = \tfrac{9}{2} = 4 \tfrac{1}{2}$
- e. $3 \tfrac{3}{4} = \tfrac{15}{4}, \; 4 \tfrac{3}{5} = \tfrac{23}{5}$ $\tfrac{15}{4} \times \tfrac{23}{5} = \tfrac{345}{20} = \tfrac{69}{4} = 17 \tfrac{1}{4}$
- f. $4 \tfrac{4}{7} = \tfrac{32}{7}, \; 2 \tfrac{5}{16} = \tfrac{37}{16}$ $\tfrac{32}{7} \times \tfrac{37}{16} = \tfrac{1184}{112} = \tfrac{148}{14} = \tfrac{74}{7} = 10 \tfrac{4}{7}$
🧠 Think like a Mathematician
Question 22:
Marcus says:
“If I multiply any positive number by a proper fraction, the answer will always be smaller than the original number.”
a) Use specialising (examples) to show that Marcus is correct.
b) Complete the general statements:
- When you multiply any positive number by an improper fraction, the answer will always be larger than the original number.
- When you multiply any positive number by a mixed number, the answer will always be larger than the original number.
c) Discuss your answers to parts a and b with other learners in your class.
🔎 Reasoning Tip
Specialising with examples: You can test cases by trying examples such as \( 8 \times \tfrac{1}{2} \), \( 4 \tfrac{1}{2} \times \tfrac{2}{3} \), \( \tfrac{5}{9} \times \tfrac{3}{10} \), etc.
👀 show answer
a) Example: $10 \times \dfrac{3}{4} = 7.5$ which is smaller than $10$. Example: $20 \times \dfrac{2}{5} = 8$, which is smaller than $20$. This confirms Marcus is correct.
b) i) With an improper fraction (e.g. $\dfrac{7}{3}$): $6 \times \dfrac{7}{3} = 14$, which is larger than $6$. ii) With a mixed number (e.g. $2\dfrac{1}{2}$): $8 \times 2\dfrac{1}{2} = 20$, which is larger than $8$. ✅ Conclusion: Proper fractions make numbers smaller; improper fractions or mixed numbers make numbers larger.
❓ EXERCISES
23. Work out these calculations. Before you do each calculation, write down if the answer should be bigger or smaller than the first number in the calculation.
a. $3 \times \tfrac{3}{4}$
b. $6 \tfrac{2}{3} \times 1 \tfrac{1}{4}$
c. $\tfrac{4}{3} \times \tfrac{9}{8}$
👀 Show answer
Answers (Q10):
- a. Prediction: smaller than $3$. $3 \times \tfrac{3}{4}=\tfrac{9}{4}=2 \tfrac{1}{4}$.
- b. Prediction: bigger than $6 \tfrac{2}{3}$. $6 \tfrac{2}{3}\times 1 \tfrac{1}{4}=\tfrac{20}{3}\times \tfrac{5}{4}=\tfrac{25}{3}=8 \tfrac{1}{3}$.
- c. Prediction: bigger than $\tfrac{4}{3}$. $\tfrac{4}{3}\times \tfrac{9}{8}=\tfrac{3}{2}=1 \tfrac{1}{2}$.
24. Here are six calculation cards.
🔎 Reasoning Tip
Describing groups: “Describe the characteristics of the groups” means explaining why you placed the calculations into the groups you did.
A $\;(\tfrac{2}{3} \tfrac{3}{4})\times \tfrac{4}{11}$ B $\;(\tfrac{3}{2}-\tfrac{7}{8})\times \tfrac{26}{15}$ C $\;3 \tfrac{1}{2}\times \tfrac{4}{5}\times \tfrac{10}{3}$
D $\; \bigl(2-1 \tfrac{1}{3}\bigr)\times \bigl(1-\tfrac{4}{5}\bigr)$ E $\;(\tfrac{5}{4})^{2}-1 \tfrac{3}{8}\times \tfrac{2}{11}$ F $\;5 \tfrac{1}{3}-5 \tfrac{3}{4}\times \tfrac{8}{9}$
a. Work out the answers to the calculations.
b. Sort the cards into groups. Describe the characteristics of the groups you have chosen.
c. Sort the cards again but this time into different groups. Describe the characteristics of the new groups you have chosen.
👀 Show answer
Answers (Q11a):
- A $=\;(\tfrac{2}{3} \tfrac{3}{4})\times \tfrac{4}{11}=\tfrac{17}{12}\times \tfrac{4}{11}=\tfrac{17}{33}$
- B $=\;(\tfrac{3}{2}-\tfrac{7}{8})\times \tfrac{26}{15}=\tfrac{5}{8}\times \tfrac{26}{15}=\tfrac{13}{12}=1 \tfrac{1}{12}$
- C $=\;3 \tfrac{1}{2}\times \tfrac{4}{5}\times \tfrac{10}{3}=\tfrac{7}{2}\times \tfrac{8}{3}=\tfrac{28}{3}=9 \tfrac{1}{3}$
- D $=\;(2-1 \tfrac{1}{3})\times (1-\tfrac{4}{5})=\tfrac{2}{3}\times \tfrac{1}{5}=\tfrac{2}{15}$
- E $=\;(\tfrac{5}{4})^{2}-1 \tfrac{3}{8}\times \tfrac{2}{11}=\tfrac{25}{16}-\tfrac{1}{4}=\tfrac{21}{16}=1 \tfrac{5}{16}$
- F $=\;5 \tfrac{1}{3}-5 \tfrac{3}{4}\times \tfrac{8}{9}=\tfrac{16}{3}-\tfrac{46}{9}=\tfrac{2}{9}$
One way to group them (Q11b):
- Results less than $1$: A $(\tfrac{17}{33})$, D $(\tfrac{2}{15})$, F $(\tfrac{2}{9})$
- Results greater than $1$: B $(1 \tfrac{1}{12})$, C $(9 \tfrac{1}{3})$, E $(1 \tfrac{5}{16})$
A different grouping (Q11c):
- Includes addition/subtraction inside brackets or powers: A, B, D, E
- Pure multiplication chain (no brackets with $ $/$-$): C, F
Be Careful
Do not forget to simplify fractions after multiplying. Cancelling common factors before multiplying can make your calculation easier and reduce mistakes.